[Paper Review] Derived Quot schemes

Ionuţ Ciocan-Fontanine, Mikhail Kapranov|ArXiv.org|May 27, 1999

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper introduces a derived analog of Grothendieck's Quot scheme, called RQuot_h(F), which is a differential graded manifold parametrizing subsheaves of a coherent sheaf F on a smooth projective variety X. The construction resolves singularities inherent in the classical Quot scheme by encoding deformation theory via a cochain complex quasiisomorphic to RHom(K, F/K), ensuring the derived object is always smooth in a homotopical sense.

ABSTRACT

Realizing a part of the Derived Deformation Theory program, we construct a "derived" analog of the Grothendieck's Quot scheme parametrizing subsheaves in a given coherent sheaf F on a smooth projective variety X. This analog is a differential graded manifold RQuot_h(F) (so it is always smooth in an appropriate sense) whose tangent space at a point represented by a subsheaf K in F, is a cochain complex quasiisomorphic to RHom(K, F/K).

Motivation & Objective

• To develop a derived enhancement of Grothendieck's classical Quot scheme that resolves its singularities.
• To realize a key component of the Derived Deformation Theory program by constructing a homotopically coherent moduli space.
• To provide a smooth, differential graded manifold structure on the moduli of subsheaves of a coherent sheaf F on a smooth projective variety X.
• To ensure the tangent complex at each point is quasiisomorphic to RHom(K, F/K), capturing the correct derived deformation theory.
• To generalize the classical Quot scheme into a derived geometric object that is always smooth in the appropriate homotopical sense.

Proposed method

• Construct a differential graded manifold RQuot_h(F) as a derived enhancement of the classical Quot scheme.
• Use the derived Hom complex RHom(K, F/K) as the tangent complex at a point corresponding to a subsheaf K ⊂ F.
• Ensure the tangent space at each point is quasiisomorphic to RHom(K, F/K), encoding derived deformation data.
• Work within the framework of derived algebraic geometry, using dg-manifolds to replace singular classical moduli spaces.
• Apply techniques from derived deformation theory to ensure the resulting object is smooth and well-behaved in the derived category.
• Utilize the formalism of derived stacks and quasi-coherent sheaves on derived schemes to define the moduli problem.

Experimental results

Research questions

• RQ1 How can one construct a derived version of the Quot scheme that is smooth in a homotopical sense?
• RQ2 What is the correct derived tangent complex for the moduli of subsheaves of a coherent sheaf F?
• RQ3 Can the classical Quot scheme be replaced by a differential graded manifold that captures derived deformation theory?
• RQ4 How does the derived Quot scheme relate to the classical Quot scheme in terms of moduli and geometry?
• RQ5 What is the role of RHom(K, F/K) in defining the derived structure of the Quot scheme?

Key findings

• The derived Quot scheme RQuot_h(F) is constructed as a differential graded manifold, ensuring it is smooth in the derived sense.
• The tangent complex at a point corresponding to a subsheaf K ⊂ F is quasiisomorphic to RHom(K, F/K), correctly encoding derived deformation data.
• The construction realizes a key goal of the Derived Deformation Theory program by providing a derived moduli space for subsheaves.
• The derived Quot scheme resolves singularities present in the classical Quot scheme through its dg-structure.
• The object RQuot_h(F) provides a well-defined, smooth moduli space for the derived category of coherent sheaves on X.
• The method establishes a framework for constructing derived moduli spaces using RHom complexes as tangent data.